$xag(x)$

We note that this type of limit can be evaluated for certain functions by dividing numerator and denominator by the highest power of $x$ that oceurs in the denominator. However, we given the following limit$,$ where this method does not work.

$\underset{x-}{lim}\frac{(ln(x){)}^2}{6x}$

For the evaluation of indeterminate forms of type $\frac{}{}$ where other methods do not work, we use L'Hospital's Rule. This rule states that if fand $g$ are differentiable and $g(x),0$ on an open interval $t$ that contains a $($except possibly at a$)$ where $\underset{xa}{lim}f(x)=10$ and $\underset{xa}{lim}g(x)=t,$ then the following holds as long as the limit on the right side exists $($or is an or $-).$

$\underset{xa}{lim}\frac{f(x)}{g(x)}=\underset{xa}{lim}\frac{f^{\text{'}}(x)}{g^{\text{'}}(x)}$

We can apply L'Hospital's Rule for the given function using $a=,f(x)=(ln(x){)}^2,$ and $g(x)=6x.$ We need to find first both $f^{\text{'}}(x)$ and $g^{\text{'}}(x).$

$f^{\text{'}}(x)=$

$g^{\text{'}}(x)=$